I. Field of the Invention
The present invention relates to digital signal processing. More particularly, the present invention relates to a novel and improved method and apparatus for adaptive filtering.
II. Description of the Related Art
Adaptive filters have been widely used in communications systems, control systems and various other systems in which the statistical characteristics of the signals to be filtered are either unknown a priori or, in some cases, slowly time-variant (nonstationary signals). Some applications of adaptive filters include adaptive antenna systems in which adaptive filters are used for beam steering and for providing null in the beam pattern to remove undesired interference, digital communication receivers in which adaptive filters are used to provide equalization of intersymbol interference and for channel identification, adaptive noise canceling techniques in which an adaptive filter is used to estimate and eliminate a noise component in some desired signal and system modeling in which an adaptive filter is used as a model to estimate the characteristics of an unknown system.
Although both infinite impulse response (IIR) and finite impulse response (FIR) filters have been considered for adaptive filtering, the FIR filter is by far the most practical and widely used. The reason for this preference is quite simple. The FIR filter has only adjustable zeroes and hence it is free of stability problems associated with adaptive IIR filters that have adjustable poles as well as zeroes. This does not mean, however, that adaptive FIR filters are always stable. On the contrary, the stability of the filter depends critically upon the algorithm used for adjusting its coefficients.
An important consideration in the use of an adaptive filter is the criterion for optimizing the adjustable filter parameters. The criterion must not only provided a meaningful measure of filter performance, but it must also result in a practically realizable algorithm.
For example, a desirable performance index in a digital communication system is the average probability of error. Consequently, in implementing an adaptive equalizer, a method wherein the selection of equalizer coefficients to minimize the average probability of error as the basis for the optimization criterion may be best suited. Unfortunately, however, the performance index (average probability of error) for this criterion is a highly nonlinear function of the filter coefficients. Although a number of algorithms are known for finding a minimum or maximum of a nonlinear function of several variables, such algorithms are unsuitable for adaptive filtering primarily because the signal statistical characteristics are unknown and possibly time variant.
In some cases a performance index that is a nonlinear function of the filter parameters possesses many relative minima (or maxima), so that one is not certain whether the adaptive filter has converged to the optimum solution or to one of the relative minima (or maxima). For these reasons, some desirable performance indices, such as the average probability of error in a digital communication system, must be rejected of the grounds that they are impractical to implement.
One criterion that provides a good measure of performance in adaptive filtering applications is the least-squares criterion, and its counterpart in a statistical formulation of the problem, namely, the mean-square-error (MSE) criterion. The least-squares (and the MSE) criterion result in a quadratic performance index as a function of the filter coefficients and hence it possesses a single minimum. The resulting algorithms for adjusting the coefficients of the filter are relatively easy to implement.